The Remodeling of Fiber Distributions in Biological Tissues: Rotation without Rotation

Abstract

Collagen remodeling in living tissues exhibits anisotropic orientation patterns commonly described by Von Mises distributions, yet the physical origin of such nonequilibrium organization remains unresolved. In the present work, we demonstrate analytically that the combined action of Malthusian growth dynamics and the introduction of linear relations governing mechanical remodeling naturally gives rise to generalized bimodal Von Mises distributions as emergent states of living matter. The theory reveals a {\it rotation without rotation} mechanism, in which fibers progressively reorient in the absence of angular mechanical coupling via selective deposition and removal along preferred directions. The resulting analytical solutions quantitatively reproduce experimentally observed distributions and establish a direct mechanobiological origin for directional statistics in biological tissues. By interpreting the evolving normalized fiber density as a probability distribution function, we formulate a dynamical Shannon entropy framework that captures the temporal emergence of microstructural organization. The theory further yields closed-form expressions for the drift of the associated Fokker--Planck equation, enabling the corresponding stochastic differential equation to be derived, thus revealing that tissue remodeling is the collective outcome of noisy single-fiber dynamics. These results establish a minimal theoretical framework that connects biomechanics, stochastic processes, and nonequilibrium statistical organization in living matter.